| L(s) = 1 | + (−3.46 + 1.12i)2-s + (−11.0 + 24.6i)3-s + (−41.0 + 29.8i)4-s + (207. + 67.4i)5-s + (10.6 − 97.8i)6-s + (−309. + 224. i)7-s + (245. − 338. i)8-s + (−484. − 544. i)9-s − 794.·10-s + (−1.31e3 + 202. i)11-s + (−280. − 1.34e3i)12-s + (−49.3 − 152. i)13-s + (819. − 1.12e3i)14-s + (−3.95e3 + 4.36e3i)15-s + (532. − 1.63e3i)16-s + (2.81e3 + 914. i)17-s + ⋯ |
| L(s) = 1 | + (−0.433 + 0.140i)2-s + (−0.409 + 0.912i)3-s + (−0.641 + 0.465i)4-s + (1.65 + 0.539i)5-s + (0.0490 − 0.452i)6-s + (−0.901 + 0.655i)7-s + (0.479 − 0.660i)8-s + (−0.664 − 0.747i)9-s − 0.794·10-s + (−0.988 + 0.152i)11-s + (−0.162 − 0.775i)12-s + (−0.0224 − 0.0691i)13-s + (0.298 − 0.410i)14-s + (−1.17 + 1.29i)15-s + (0.129 − 0.399i)16-s + (0.572 + 0.186i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.951 + 0.308i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.951 + 0.308i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(0.0966410 - 0.611507i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0966410 - 0.611507i\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (11.0 - 24.6i)T \) |
| 11 | \( 1 + (1.31e3 - 202. i)T \) |
| good | 2 | \( 1 + (3.46 - 1.12i)T + (51.7 - 37.6i)T^{2} \) |
| 5 | \( 1 + (-207. - 67.4i)T + (1.26e4 + 9.18e3i)T^{2} \) |
| 7 | \( 1 + (309. - 224. i)T + (3.63e4 - 1.11e5i)T^{2} \) |
| 13 | \( 1 + (49.3 + 152. i)T + (-3.90e6 + 2.83e6i)T^{2} \) |
| 17 | \( 1 + (-2.81e3 - 914. i)T + (1.95e7 + 1.41e7i)T^{2} \) |
| 19 | \( 1 + (8.90e3 + 6.47e3i)T + (1.45e7 + 4.47e7i)T^{2} \) |
| 23 | \( 1 - 3.86e3iT - 1.48e8T^{2} \) |
| 29 | \( 1 + (650. + 895. i)T + (-1.83e8 + 5.65e8i)T^{2} \) |
| 31 | \( 1 + (-9.41e3 - 2.89e4i)T + (-7.18e8 + 5.21e8i)T^{2} \) |
| 37 | \( 1 + (3.22e4 - 2.34e4i)T + (7.92e8 - 2.44e9i)T^{2} \) |
| 41 | \( 1 + (-3.52e3 + 4.85e3i)T + (-1.46e9 - 4.51e9i)T^{2} \) |
| 43 | \( 1 + 8.88e4T + 6.32e9T^{2} \) |
| 47 | \( 1 + (6.28e4 - 8.65e4i)T + (-3.33e9 - 1.02e10i)T^{2} \) |
| 53 | \( 1 + (-1.55e4 + 5.04e3i)T + (1.79e10 - 1.30e10i)T^{2} \) |
| 59 | \( 1 + (-3.17e4 - 4.37e4i)T + (-1.30e10 + 4.01e10i)T^{2} \) |
| 61 | \( 1 + (1.15e5 - 3.54e5i)T + (-4.16e10 - 3.02e10i)T^{2} \) |
| 67 | \( 1 - 3.72e5T + 9.04e10T^{2} \) |
| 71 | \( 1 + (6.48e4 + 2.10e4i)T + (1.03e11 + 7.52e10i)T^{2} \) |
| 73 | \( 1 + (-4.12e4 + 2.99e4i)T + (4.67e10 - 1.43e11i)T^{2} \) |
| 79 | \( 1 + (2.24e5 + 6.90e5i)T + (-1.96e11 + 1.42e11i)T^{2} \) |
| 83 | \( 1 + (-6.96e5 - 2.26e5i)T + (2.64e11 + 1.92e11i)T^{2} \) |
| 89 | \( 1 - 1.34e6iT - 4.96e11T^{2} \) |
| 97 | \( 1 + (1.96e5 + 6.04e5i)T + (-6.73e11 + 4.89e11i)T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−16.30882016905690354397406895951, −15.06083202611986000082627834929, −13.60465523221922694167992798578, −12.62523472273710722677464305765, −10.49040245048700678514074597473, −9.788406549195820887197875174313, −8.816209868340149431191330864155, −6.46926267472601525122669147914, −5.16187332277954643448191647663, −2.93842939646340985249684819140,
0.38308269572734344762138452929, 1.92434619963231598257759868099, 5.28186322476989606416708976539, 6.34187026607618866011449638968, 8.310899564704601167258013560321, 9.802246063738534201611443545463, 10.52951716304319101556478670851, 12.79043640414195980412320224883, 13.34527408409747809884298817149, 14.24407519484033289769041658647
